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Physical Review B

Coherence and antibunching in a trapped interacting Bose-Einstein condensate

Sheng-Jun Yang, Sajeev John

Published Jul 13, 2011• 7 citations
Publisher

Abstract

We present a model for the equilibrium quantum statistics of a condensate of repulsively interacting bosons in a two-dimensional trap. Particle correlations in the ground state are treated exactly, whereas interactions with excited particles are treated in a generalized Bogoliubov mean-field theory. This leads to a fundamental physical picture for the condensation of interacting bosons through an anharmonic oscillator ground state coupled to excited Bogoliubov quasiparticles in which the quantum number statistics of condensate particles emerges self-consistently. Below the Bose-Einstein condensation temperature, our model exhibits a crossover from particle bunching to Poissonian statistics and finally antibunching as the temperature is lowered or as the trapping area is decreased. When applied to Bose condensation of long-lived dressed excitons in a photonic band gap material, our model suggests that this system may serve as a novel tunable source for nonclassical states of light.

DOI
10.1103/physrevb.84.024515

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We present a model for the equilibrium quantum statistics of a condensate of repulsively interacting bosons in a two-dimensional trap. Particle correlations in the ground state are treated exactly, whereas interactions with excited particles are treated in a generalized Bogoliubov mean-field theory. This leads to a fundamental physical picture for the condensation of interacting bosons through an anharmonic oscillator ground state coupled to excited Bogoliubov quasiparticles in which the quantum number statistics of condensate particles emerges self-consistently. Below the Bose-Einstein condensation temperature, our model exhibits a crossover from particle bunching to Poissonian statistics and finally antibunching as the temperature is lowered or as the trapping area is decreased. When applied to Bose condensation of long-lived dressed excitons in a photonic band gap material, our model suggests that this system may serve as a novel tunable source for nonclassical states of light.
We demonstrate electromagnetically-induced anomalous quantum dynamics of an exciton in a photonic band gap (PBG) - quantum well (QW) hetero-structure. Within the engineered electromagnetic vacuum of the PBG material, the exciton can propagate through the QW by the emission and re-absorption of virtual photons in addition to the conventional electronic hopping mechanism. When the exciton wavevector and recombination energy coincide nearly with a photonic band edge, the exciton kinetic energy is lowered by 1-10meV through coherent radiative hopping. This capture of the exciton by the photonic band edge is accompanied by strong electromagnetic dressing in which the exciton's renormalized effective mass is 4-5 orders of magnitude smaller than in the absence of the PBG environment. This dressed exciton exhibits a long radiative lifetime characteristic of a photon-atom bound state and is robust to phonon-assisted, re-combinative decay. By inheriting properties of the PBG electromagnetic vacuum, the bound electron-hole pair becomes a stable, ultra-mobile quantum excitation. Unlike traditional exciton-polariton modes created by placing a QW in a one-dimensional optical cavity, our PBG-QW excitons exhibit strong coupling to optical modes and retain a long lifetime. This is crucial for unambiguous observation of quantum coherence effects such as Bose-Einstein condensation. We present a model for the equilibrium quantum statistics of a condensate of repulsively interacting bosons in a two-dimensional trap. Particle correlations in the ground state are treated exactly, whereas interactions with excited particles are treated in a generalized Bogoliubov mean-field theory. This leads to a fundamental physical picture for condensation of interacting bosons through an anharmonic oscillator ground state coupled to excited Bogoliubov quasiparticles in which the quantum number statistics of condensate particles emerges self-consistently. Our anharmonic oscillator model for the exciton ground state manifold goes beyond the conceptual framework of traditional Bogoliubov theory. Below the Bose-Einstein condensation temperature, our model exhibits a crossover from particle bunching to Poissonian statistics and finally antibunching as temperature is lowered or as the trapping area is decreased. When applied to Bose condensation of long-lived dressed excitons in a photonic band gap material, our model suggests that this system may serve as a novel tunable source for non-classical states of light.
We consider equilibrium canonical-ensemble fluctuations of the Bose-Einstein condensate (BEC) in a trap using the particle-number conserving operator formalism of Girardeau and Arnowitt [Phys. Rev. 113, 755 (1959)] and the concept of the canonical-ensemble quasiparticles. We find explicit expressions for the characteristic function and all cumulants of the ground-state occupation statistics both for the dilute weakly interacting and ideal Bose gases. We present analytical formulas for the moments of the ground-state occupation fluctuations in the weakly interacting and ideal Bose gases in a box (``homogeneous gas'') as well as in the ideal Bose gas in an arbitrary power-law trap, in particular, in an arbitrary harmonic trap. We calculate the effect of Bogoliubov coupling between quasiparticles on suppression of the ground-state occupation fluctuations at moderate temperatures and their enhancement at very low temperatures. We clarify a crossover between ideal-gas and weakly interacting gas statistics, which is governed by a pair-correlation, squeezing mechanism. The important conclusion is that in most cases the ground-state occupation fluctuations are anomalously large and are not Gaussian even in the thermodynamic limit.
We consider a Bose-Einstein condensate in a double-well potential undergoing a dynamical transition from the regime of Josephson oscillations to the regime of self-trapping. We analyze the statistical properties of the ground state (or the highest excited state) of the Hamiltonian in these two regimes for attractive (repulsive) interactions. We demonstrate that it is impossible to describe the transition within the mean-field theory. In contrast, the transition proceeds through a strongly correlated delocalized state, with large quantum fluctuations, and spontaneous breaking of the symmetry.
We analyze the mean-field equilibrium states of a homogeneous Bose-Einstein condensate of boson particles in different internal states, possibly in the presence of an internal Josephson interaction, and we describe the quantum fluctuations by constructing an explicit form of the Bogoliubov transformation that characterizes the quasiparticle excitations of the system. The derivation of the Bogoliubov transformation is based on a simple geometrical picture which leads to a convenient parametrization of the transformation. Our calculations of physical properties, such as depletions and phonon velocities, as well as our discussion of stability, indicate the sense and the extent to which the internal Josephson interaction controls the many-body physics of multicomponent phase coherent matter.
Summary form only given.We use quantum kinetic theory to calculate the steady-state and the fluctuations of a trapped Bose-Einstein condensate at finite temperature. We divide the system into two parts. The first part consists of the condensate particles. They are described quantum mechanically using the number-conserving Bogoliubov method that allows us to characterize the condensate by the number of condensed particles. The second part are the noncondensed particles. They are treated as a classical gas in thermal equilibrium. Using quantum kinetic theory we find a master equation for the reduced density operator of the Bose-Einstein condensate. We calculate the steady-state of the system and investigate the effect of one-, two- and three-particle losses on the condensate. Using linearized Ito equations we find expressions for the intensity fluctuations and the amplitude fluctuations in the condensate. The effect of the trap losses on the particle statistics is also studied. We then assume that the condensate particles are coherently pumped into an untrapped internal energy level and fall down in gravity. For this situation we derive the relation between the statistics of the trapped condensate particles and the statistics of the particles arriving at an atom detector positioned below the condensate.
We revisit the dynamics of a Bose-Einstein condensate in a double-well potential, from the regime of Josephson plasma oscillations to the self-trapping regime, by means of the Bogoliubov quasiparticle projection method. For a very small imbalance between the left and right wells only the lowest Bogoliubov mode is significantly occupied. In this regime the system performs plasma oscillations at the corresponding frequency, and the evolution of the condensate is characterized by a periodic transfer of population between the ground and the first excited state. As the initial imbalance is increased, more excited modes---though initially not macroscopically occupied---get coupled during the evolution of the system. Since their population also varies with time, the frequency spectrum of the imbalance turns out to be still peaked around a single frequency, which is continuously shifted towards lower values. The nonlinear mixing between Bogoliubov modes eventually drives the system into the the self-trapping regime, when the population of the ground state can be transferred completely to the excited states at some time during the evolution. For simplicity, here we consider a one-dimensional setup, but the results are expected to hold also in higher dimensions.